### Brain Teasers

# Cafeteria Capacity

The Madison School has a new cafeteria with very artistic, new tables for its 375 students. Besides the big round tables, some tables are octagons, pentagons, hexagons or squares. The students eat lunch in 3 shifts, with an equal number of students dining during each lunch period. Because the cafeteria is new, they've allowed extra space in case there are more students in the future. Right now, there are 47 empty seats during each lunch period.

Each round table seats ten students. Each of the other types of tables can seat one student on each side.

There are at least 4 tables of each shape.

The same total number of students can be seated at the square tables as at the pentagons, but there are more square tables than pentagons.

The total number of students who can be seated at round tables is equal to the total number of students at the square and pentagon shaped tables combined.

There are more octagonal tables than any other shape.

There is a total of 4 fewer places at the hexagons than at the round tables.

How many tables are there of each shape?

Each round table seats ten students. Each of the other types of tables can seat one student on each side.

There are at least 4 tables of each shape.

The same total number of students can be seated at the square tables as at the pentagons, but there are more square tables than pentagons.

The total number of students who can be seated at round tables is equal to the total number of students at the square and pentagon shaped tables combined.

There are more octagonal tables than any other shape.

There is a total of 4 fewer places at the hexagons than at the round tables.

How many tables are there of each shape?

### Hint

Make a chart, and then guess and check to calculate the total number of places at each type of table.### Answer

There are 4 round tables, 7 octagonal tables, 6 hexagonal tables, 4 pentagonal tables, and 5 square tables.Table Shape Number of Tables Number of Seats Total Seats

square 5 4 20

pentagon 4 5 20

hexagon 6 6 36

octagon 7 8 56

round 4 10 40

Total places 172

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